UNIVERSIT `
A DEGLI STUDI
DI MODENA E REGGIO EMILIA
Dottorato di ricerca in Matematica in convenzione con l’Universit`a degli Studi di Ferrara e l’Universit`a degli Studi di Parma
Ciclo XXXII
ON PLANE CREMONA MAPS OF SMALL DEGREE AND
THEIR QUADRATIC LENGTHS
Candidato: NGUYEN THI. NGO˜ˆ . C GIAO
Relatore: Prof. ALBERTO CALABRI
Coordinatore del Corso di Dottorato: Prof. CRISTIAN GIARDIN `A
Abstract in English and Sintesi in italiano
On plane Cremona maps of small degree and their
quadratic lengths
Abstract. The plane Cremona group Cr(P2) is the group of birational transformations of the complex projective plane. By the famous Noether-Castelnuovo theorem, every birational map ϕ ∈ Cr(P2) is the composition of finitely many (ordinary) quadratic maps. This leads
to the notion of (ordinary) quadratic length of a given plane Cremona map. While quadratic maps are classically very well-known, only recently Cerveau and D´eserti extensively studied and gave a classification of cubic plane Cremona maps. However, it turns out that their classification is not complete and it contains some inaccuracies.
In this thesis, we first give a fine and complete classification of cubic plane Cremona maps, up to a natural notion of equivalence, by using the so-called enriched weighted proximity graph associated to the base points of the homaloidal net defining the given cubic plane Cremona map. We then classify such enriched weighted proximity graphs also for quartic plane Cremona maps. This allows to compute exactly the ordinary quadratic length and the quadratic length of cubic plane Cremona maps and, in many cases, also of quartic plane Cremona maps.
Sulle trasformazioni piane di Cremona di grado
basso e le loro lunghezze quadratiche
Sintesi. Il gruppo di Cremona Cr(P2) `e il gruppo di trasformazioni birazionali del pi-ano proiettivo complesso. Per il famoso teorema di Noether-Castelnuovo, ogni trasfor-mazione birazionale ϕ ∈ Cr(P2) `e la composizione di un numero finito di trasformazioni quadratiche (ordinarie). Ci`o porta alla nozione di lunghezza quadratica (ordinaria) di una data trasformazione cremoniana. Mentre le trasformazioni quadratiche sono classicamente molto conosciute, solo recentemente Cerveau e D´eserti hanno studiato in dettaglio e dato una classificazione delle trasformazioni cremoniane cubiche. Tuttavia, `e risultato che la loro classificazione `e incompleta e contiene qualche inaccuratezza.
In questa tesi, prima diamo una classificazione fine e completa della trasformazioni cremo-niane cubiche, a meno di una nozione naturale di equivalenza, usando il cosiddetto grafo di prossimit`a pesato e arricchito, associato ai punti base della rete omaloidica che definisce la data trasformazione cremoniana cubica. Poi classifichiamo tali grafi di prossimit`a pesati e arricchiti anche per le trasformazioni cremoniane quartiche. Ci`o ci permette di calcolare esattamente le lunghezze quadratiche (ordinarie) delle trasformazioni cremoniane cubiche e, in molti casi, anche di quelle quartiche.
Keywords and phrases: cubic plane Cremona maps, quartic plane Cremona maps, quadratic length, ordinary quadratic length, de Jonqui`eres maps.
Acknowledgements
I would like to thank all those people who made this thesis possible and unforgettable expe-rience for me.
I would like to express my gratitude to my supervisor Prof. Alberto Calabri, who helped me moving my first steps in studying Birational Geometry and coming up with this thesis topic. He has always supported my work through helpful advices, enjoyable conversations and dedicated guides. Thank you for your help, your ideas and the discussions we have had throughout these years.
Special thanks to Prof. Massimiliano Mella, Prof. Paltin Ionescu and Prof. Ellia Filippo Alfredo at Universit`a degli Studi di Ferrara for all attractive lectures, seminars and great supports. Many thanks to the Secretary’s office and Technical administrative office for their great help.
I really appreciate to Universit`a degli Studi di Modena e Reggio Emilia for having funded my Ph.D. studies. I must also thank Prof. Cristian Giardin`a, my Ph.D. Coordinator, for his help and support.
I wish to express my gratitude to the referees, Prof. Ciro Ciliberto from Universit`a di Roma Tor Vergata and Prof. J´er´emy Blanc from Universit¨at Basel, for valuable corrections and suggestions.
I am grateful to Prof. Ta Le Loi, Prof. Pham Tien Son, Prof. Nguyen Viet Dung, Prof. Dang Tuan Hiep and all other professors at Dalat University and Hanoi Institute of Math-ematics, who first encouraged me to pursue scientific research. I would also like to thank Prof. Fr´ed´eric Patras and Prof. Andreas H¨oring both from Laboratoire J.A. Dieudonn´e, who suggested and introduced me to enroll for a Ph.D. position in Italy. Thanks to all for their greetings and encouragement throughout these years.
I am thankful for the friendship of all the great people that I have met throughout my years in Ferrara. To my friends Elisa Maragliotta, Federica Semeraro, Serena Crisci, Daniele
Acerra, Giorgio Menegatti, Francesco Galuppi, Sara Durighetto, Leonardo Spinosa, Alex Casarotti, Sara Bagossi, Elsa Corniani, Matteo Bergami, Imelda Shehu, Ha T. Nguyen, Son T. Pham, Ngoc N. Dao and many other people, thank you for sharing aperitivo hours, Italian cuisine, Italian culture, dreams, the joys and also the sadness. I also thank my friends back home, specially to Nguyen T.N. Bui, Tien T.T. Vo, Quy M. Pham, Chau T.M. Le, TNK33, CHTGTK21 and many other, chatting with them help me less homesick. I could not have done this without you.
I am deeply indebted to my parents, two my sisters and my big family. They have cherished with me every great moment and helped me right when I was in need. Their constant love and encouragement has always been with me. Thank you for being there for me.
Introduction
We work over the field C of complex numbers.
We denote by P2 the projective plane and by Cr(P2) the plane Cremona group, that is the group of birational maps P2 99K P2. Recall that the celebrated Noether-Castelnuovo Theorem says that Cr(P2) is generated by the automorphisms of P2 and the elementary quadratic transformation
σ : P2 99K P2, [x : y : z] 7→ [yz : xz : xy].
Note that a presentation of Cr(P2) involving exactly these generators have been found only very recently by Urech and Zimmermann in [24].
In other words, any plane Cremona map ϕ : P2 99K P2 can be written as
ϕ = αn◦ σ ◦ αn−1◦ σ ◦ · · · ◦ α1 ◦ σ ◦ α0
where αi ∈ Aut(P2) for any i = 0, . . . , n, for some integer n.
Let us say that a decomposition of ϕ as above is “minimal” if so is n among all decomposi-tions of ϕ. Let us call such n the “ordinary quadratic length” of ϕ and denote it by oql(ϕ). Recall that a quadratic plane Cremona map is called “ordinary” if it has three proper base points. In other words, oql(ϕ) is the minimal number of ordinary quadratic maps needed to decompose ϕ.
Similarly, let us define the “quadratic length” of a plane Cremona map ϕ as the minimal number of quadratic maps needed to decompose ϕ and let us denote it by ql(ϕ).
Let us say that two plane Cremona maps ϕ, ϕ0: P2 99K P2 are equivalent if there exist two
automorphisms α, α0 ∈ Aut(P2) such that ϕ0 = α0 ◦ ϕ ◦ α. The classification of equivalence
classes of quadratic plane Cremona maps is very well-known from the beginning of the study of plane Cremona maps more than one hundred years ago.
Nonetheless, a classification of equivalence classes of cubic plane Cremona maps has been described only few years ago by Cerveau and D´eserti in [11]: they find 32 types of cubic
plane Cremona maps, namely 27 types are a single map whereas 4 types are families of maps depending on one parameter and one type is a family of maps depending on two parameters. Their classification is based on the detailed analysis of those plane curves which are contracted by a cubic plane Cremona map.
However, it turns out that the classification in [11] is not complete and it contains some inaccuracies, see Section 4.2 for a more detailed account:
• we found a map that does not occur in their list;
• we found that their type 17, that is a single map, should be replaced by a one-parameter set of maps;
• we found that their type 19 is equivalent to a particular case of their type 18; • we found that their type 31 is equivalent to a particular case of their type 30.
One of the main purpose of this thesis is giving a complete classification of equivalence classes of cubic plane Cremona maps. Our classification is based on the study of enriched weighted proximity graphs of the base points of the homaloidal net defining a plane Cremona map. Accordingly, we divide cubic plane Cremona maps into 31 types, namely 25 types are single maps, 5 types are families of maps depending on one parameter and 1 type is a family of maps depending on two parameters. Two maps of two different types are not equivalent. Moreover, we find the conditions when two maps of the same type (depending on parame-ters) are equivalent. Then, using our classification, we compute exactly the quadratic length and ordinary quadratic length of all cubic plane Cremona maps.
Furthermore, we generalize this approach to study quartic plane Cremona maps and we compute their quadratic length and ordinary quadratic length. Concerning quartic plane Cremona maps, recall that they can divided in de Jonqui`eres maps, that have a triple base point and 6 simple base points, and non-de Jonqui`eres maps, that have 3 double base points and 3 simple base points. We give a complete list of all possible enriched weighted proximity graphs of the base points of all quartic plane Cremona maps, namely there are exactly 449 types of enriched weighted proximity graphs of quartic de Jonqui`eres maps and 119 types of enriched weighted proximity graphs of quartic non-de Jonqui`eres maps. Using these clas-sifications, we compute the quadratic lengths and the ordinary quadratic lengths of many quartic Cremona maps.
In details, this thesis is divided into five chapters.
In Chapter 1: a very brief summary of the most relevant results about plane curves, blowing-ups and plane birational maps is provided with little or no proof, simply to fix notation and to set the stage. In particular, we give a way to describe infinitely near points that we call standard coordinates. Some applications to plane conics are presented right after that.
Let us describe the content of Chapter 2: we recall in detail the proximity matrices and the admissible oriented graphs which encode sequences of blowing-ups. It allows us to define the so-called enriched weighted proximity graph for a given plane Cremona map, based on proximity relations among the base points of the map, together with some other properties, for instance collinearity properties of the base points or at least 6 base points are on an irreducible conic and so on.
In Chapter 3: we introduce the notion of quadratic length and ordinary quadratic length. We study their first properties, in particular those related with weighted proximity graphs of de Jonqui`eres maps.
In Chapter 4: we give a complete classification of equivalence classes of cubic plane Cremona maps. This allows us compute the quadratic length and the ordinary quadratic length of all cubic plane Cremona maps.
We finish the thesis with Chapter 5, where we extensively study quartic plane Cremona maps and their quadratic length and ordinary quadratic length.
Contents
Abstract i
Acknowledgements ii
Introduction iv
1 Generalities on plane Cremona maps 1
1.1 Plane curves . . . 1
1.2 Blowing-ups . . . 6
1.3 Standard coordinates of infinitely near points . . . 13
1.4 Plane Cremona maps . . . 19
2 Weighted proximity graphs of plane Cremona maps 24 2.1 Admissible digraphs . . . 24
2.2 Proximity matrices . . . 26
2.3 Weighted proximity graph of a plane Cremona map . . . 29
2.4 Enriched weighted proximity graph of a plane Cremona map . . . 38
3 Lengths in the Cremona group 42 3.1 Decompositions of a plane Cremona map . . . 42
3.2 Quadratic length of de Jonqui`eres maps . . . 47
3.3 On ordinary quadratic length of de Jonqui`eres maps . . . 49
4 Cubic plane Cremona maps 54 4.1 Classification theorems . . . 54
4.2 Comparison with the classification in [11] . . . 58
4.3 Proof of the Classification Theorem 4.1 . . . 63
4.4 Ordinary quadratic length of cubic plane Cremona maps . . . 81
5.1 Quartic plane de Jonqui`eres maps . . . 88 5.2 Quartic plane non-de Jonqui`eres maps . . . 121
Bibliography 128
Chapter 1
Generalities on plane Cremona maps
A comprehensive understanding of plane Cremona maps requires some background in alge-braic geometry. This chapter aims to recall basic concepts, properties and well-known facts of plane Cremona maps, simply to fix notation and to set the stage. Most results in this chapter can be found in almost any introduction to algebraic geometry and, for a more in depth treatment, we suggest some sources on the subject such as [16, 19].
Throughout this thesis, we work over C, the field of complex numbers. To avoid confusion, we adopt the following notational conventions.
Notation 1.1. Any non-zero complex number z can be written uniquely as follows z = reiθ = r cos(θ) + i sin(θ), with r > 0, and θ ∈ [0, 2π).
The angle θ is called the argument of z and the real number r is the norm of z. Any non-zero complex number z = r cos(θ) + i sin(θ) has two square roots, namely
z0 = √ r cos θ 2 + i sin θ 2 , z1 = −z0.
From now on, we denote z0 by
√
z and z1 by −
√ z. For any t ∈ C such that t2 6= 4, set t• =√t2 − 4, t
+ = (t + t•)/2 and t− = (t − t•)/2, that
is, t± are the roots of the equation x2− tx + 1 = 0. Note that, if t2 6= 4 then t+ 6= t− and
t+, t− 6= 0.
By a surface, we mean a smooth projective irreducible algebraic surface over C.
1.1
Plane curves
1.1.1
Affine curves in C
2Let f (x, y) be a non-constant polynomial in two variables with complex coefficients. One says that f (x, y) has no repeated factors if one cannot write
f (x, y) = g(x, y)2h(x, y)
where g(x, y) and h(x, y) are polynomials and g(x, y) is non-constant.
Definition 1.2. Let f (x, y) be a non-constant polynomial in two variables with complex coefficients and no repeated factors. Then, the affine curve C in C2 defined by f (x, y) is
C = (x, y) ∈ C2 f (x, y) = 0 .
Remark 1.3. Two polynomials f (x, y) and g(x, y) define the same affine curves in C2 if
and only if they are scalar multiples of each other, and a polynomial with repeated factors is then thought of as defining a curve with multiplicities attached.
Definition 1.4. The degree d of the curve C defined by f (x, y) =P
r,scr,sx
rys is the degree
of the polynomial f , i.e.
d = maxr + s
cr,s6= 0 .
Definition 1.5. An affine curve C defined by a polynomial f (x, y) is called irreducible if the polynomial f is irreducible, that is, if f (x, y) has no factors other than constants and scalar multiples of itself.
If the irreducible factors of f (x, y) are
f1(x, y), . . . , fk(x, y),
then the curves defined by fi(x, y) are called the irreducible components of C for any i =
1, . . . , k.
1.1.2
The projective plane
Definition 1.6. The complex projective plane P2 is the set C3\(0, 0, 0) ∼ where ∼ is the equivalence relation
(x, y, z) ∼ (x0, y0, z0) ⇔ ∃λ ∈ C∗ : x0 = λx, y0 = λy, z0 = λz. A point of P2 is denoted by [x : y : z].
Note that P2 is covered by three affine charts, namely P2 = U0∪ U1∪ U2 where U0 =[x : y : z] ∈ P2 x 6= 0 , U1 =[x : y : z] ∈ P2 y 6= 0 , U2 =[x : y : z] ∈ P2 z 6= 0 ,
and one can identify Ui with C2 for each i = 0, 1, 2. For instance, one has U2 ' C2 where an
isomorphism φ : U2 → C2x,y is defined by
φ([x : y : z]) = x z, y z (1.1) with inverse (x, y) 7−→ [x : y : 1].
The complement of U2 in P2 is the projective line defined by z = 0 which we can identify
with P1 via the map
[x : y : 0] 7−→ [x : y].
In other words, P2 is the disjoint union of a copy of C2 and a copy of P1 which we think of
as “the line at infinity”.
1.1.3
Projective curves in P
2Recall that a polynomial F (x, y, z) is called homogeneous of degree d if F (λx, λy, λz) = λdF (x, y, z)
for all λ ∈ C. Note that the first partial derivatives of F are homogeneous polynomials of degree d − 1.
Definition 1.7. Let F (x, y, z) be a non-constant homogeneous polynomial in three variables x, y, z with complex coefficients. Assume that F (x, y, z) has no repeated factors. Then, the projective curve C in P2 defined by F (x, y, z) is
C =[x : y : z] ∈ P2 F (x, y, z) = 0 .
Note that the condition F (x, y, z) = 0 is independent of the choice of homogeneous coordi-nates [x : y : z] because F is a homogeneous polynomial and hence
F (λx, λy, λz) = 0 ⇐⇒ F (x, y, z) = 0 for any λ ∈ C∗.
Remark 1.8. Just as for curves in C2, it is in fact that the case that two homogeneous
poly-nomials F (x, y, z) and G(x, y, z) with no repeated factors define the same projective curves in P2 if and only if they are scalar multiples of each other, and a homogeneous polynomial
with repeated factors can be thought of as defining a curve with multiplicities attached to its components.
Definition 1.9. The degree of a projective curve C in P2 defined by a homogeneous poly-nomial F (x, y, z) is the degree of F (x, y, z). The curve C is called irreducible if F (x, y, z) is irreducible, i.e. F (x, y, z) has no non-constant polynomial factors other than scalar multiples of itself. An irreducible projective curve D defined by a homogeneous polynomial G(x, y, z) is called a component of C if G(x, y, z) divides F (x, y, z).
1.1.4
From affine to projective curves and vice versa
Affine and projective curves are closely related. From an affine curve C one can obtain a projective curve ˜C by adding points at infinity. Vice versa, from a projective curve ˜C one can obtain an affine curve C by discarding points at infinity.
Let F (x, y, z) be a non-constant homogeneous polynomial of degree d. Under the identifi-cation (1.1) of U2 with C2, the intersection with U2 of the projective curve ˜C defined by
F is the affine curve C in C2 defined by the (possibly inhomogeneous) polynomial in two variables
F (x, y, 1).
This polynomial has degree d provided that z = 0 is not a factor of F (x, y, z) (i.e. ˜C does not contain the line z = 0).
Conversely, if f (x, y) is a polynomial of degree d in two variables x and y, say f (x, y) = X
r+s≤d
ar,sxrys,
then the affine curve C defined by f (x, y) is the intersection of U2 (identified with C2) with
the projective curve ˜C in P2 defined by the homogeneous polynomial zdf x z, y z = X r+s≤d ar,sxryszd−r−s.
The intersection of this projective curve with the line at infinity z = 0 is the set of points [x : y : 0] ∈ P2
X
0≤r≤d
ar,d−rxryd−r = 0 .
However, the polynomial
X
0≤r≤d
ar,d−rxryd−r
can be factorised as a product of linear factors Y
1≤i≤d
αix + βiy.
This factors correspond to points [−βi : αi] in P1; when P1 is identified with the line z = 0
in P2, these points are precisely the points of ˜C \ C.
In this way, we get a bijective correspondence between affine curves C in C2 and projective
1.1.5
Automorphisms of the projective plane
The projective plane is an excellent backdrop for studying the classical algebraic geometry, and so, among other things, it will be useful to understand automorphisms of the projective plane.
Notation 1.10. We denote by AutC(P2), or simply Aut(P2), the group of automorphisms
of P2, that it is isomorphic to the quotient P GL
3 of the general linear group GL3 by the
one-dimensional subgroup of scalar matrices λI | λ ∈ C∗ , see for instance Proposition 11.46, §11 in [18].
More precisely, an automorphism α : P2 → P2 is of the following form
α([x : y : z]) =a11x + a12y + a13z : a21x + a22y + a23z : a31x + a32y + a33z
where aij ∈ C for any i, j ∈ {1, 2, 3} and the (3 × 3)-matrix M = aij satisfies det(M) 6= 0.
One says that M is the associated matrix of the automorphism α, or simply one says that α is defined by M .
Lemma 1.11. (The Four Points Lemma) Let pi = [xi : yi : zi] (i = 1, 2, 3, 4) be four
points in the projective plane such that no three of them are collinear. Then, there is a unique automorphism of P2, sending e
1 = [1 : 0 : 0], e2 = [0 : 1 : 0], e3 = [0 : 0 : 1] and
e4 = [1 : 1 : 1], to p1, p2, p3 and p4, respectively.
Proof. See §11.2 in [17].
Definition 1.12. Two projective curves defined respectively by two polynomials F, G in P2 are called projectively equivalent if there exists an automorphism α of P2 and a scalar λ ∈ C∗, for which G = λ(F ◦ α).
Note that projective equivalence is an equivalence relation, and that projectively equivalent curves have the same degree. Moreover, F is reduced if and only if so is G.
1.1.6
Plane conics
Any conic C in P2 is defined by a quadratic polynomial
Q(x, y, z) = x y z A x y z T
where A is a (3 × 3) non-zero symmetric complex matrix.
Note that C is irreducible if and only if det(A) 6= 0.
More precisely, a plane conic C is defined as follows
which is associated to the matrix A = 1 2 2a b d b 2c e d e 2f . .
Remark 1.13. Let C be an irreducible conic and ` be a line in P2. Then, C ∩ ` is nonempty
and it consists of at most two points.
When C ∩ ` is just one point p0, one says that ` is tangent to C at p0 and we denote ` by
Tp0(C).
Note that, if p0 ∈ C, then C has a unique tangent line at p0, while, if p0 ∈ C, then there are/
exactly two tangent lines to C passing through p0.
Lemma 1.14 (cf. [23, Lem 1.2.3, Sec 1.2]). Any two irreducible conics can be mapped each other by projective transformations.
Proof. Let C be an irreducible conic. It suffices to show that there exists a projective transformation that maps C to the conic C0: xz − y2 = 0. On C, take mutually distinct
points p1, p2 and p3. Let p4 be the intersection point of Tp1(C) and Tp2(C). Clearly, no three
among p1, p2, p3, p4 are collinear. Therefore, by Lemma 1.11, there exists an automorphism
α of P2 that sends p
1, p2, p3, p4 to e1, e3, e4, e2, respectively. Hence, α sends C to the conic
C0.
The proof of the previous lemma shows also the following:
Lemma 1.15. Let n ∈ {1, 2, 3}. Let C1, C2 be irreducible conics. Let p1, . . . , pn ∈ C1 and
let q1, . . . , qn ∈ C2. Then, there exists an auotmorphism α of P2 such that α(C1) = C2 and
α(pi) = qi, i = 1, . . . , n.
We recall the following result, taken directly from §5.2 of Chapter 5 in [26]:
Lemma 1.16. Suppose p1, p2, p3, p4, p5 ∈ P2 are any five points such that no three of them
are collinear. Then, there is a unique irreducible conic passing through p1, . . . , p5.
In Section 1.3.1, we will generalize the previous result to infinitely near points, when it is possible.
1.2
Blowing-ups
The notion of blowing-up is the most fundamental one in the subject of birational geometry. In this section, we study the blowing-up map. References for this section are [3] and [19].
1.2.1
Blowing-up of a surface at a point
Firstly, we will construct the blowing-up of A2 at 0 := (0, 0). Consider the product A2× P1
, suppose that x, y are the affine coordinates of A2 and u, v are the homogeneous coordinates of P1. Then,
Definition 1.17. The blowing-up of A2 at 0 is the closed subset Bl
0(A2) of A2× P1 defined
by
Bl0(A2) :=
(x, y), [u : v] ∈ A2× P1| xv = uy .
We have a natural morphism ϕ : Bl0(A2) → A2 obtained by restricting the projection map
pr1 of A2× P1 onto the first factor. In other words, the following diagram commutes:
Bl0(A2) A2× P1
A2. pr1
ϕ
Lemma 1.18. (1) If p ∈ A2 and p 6= 0, then ϕ−1(p) consists of a single point.
(2) ϕ−1(0) ' P1.
(3) The points of ϕ−1(0) are in one-to-one correspondence with the set of lines through 0 in A2.
(4) Bl0(A2) \ ϕ−1(0) is isomorphic to A2 \ 0 .
(5) Bl0(A2) is irreducible.
Proof. (1) Let p = (x0, y0) ∈ A2 \ 0 , suppose that x0 6= 0 (resp. y0 6= 0). Now, if
p, [u : v] ∈ ϕ−1(p) then v = y0 x0 u (resp. u = x0 y0 v), so [u : v] is uniquely determined as a point in P1. By setting u = x
0 (resp. v = y0), we have [u : v] = [x0 : y0]. Thus,
ϕ−1(p) consists of a single point.
(2) ϕ−1(0) consists of all points 0, [u : v] for any [u : v] ∈ P1, subject to no restriction.
(3) A line l through 0 in A2 can be given by parametric equations
x = at, y = bt | t ∈ A1
where a, b ∈ C are not both zero. Now, consider the line l0 = ϕ−1 l \ 0 in Bl0(A2) \ ϕ−1(0). It is given parametrically by
Since u, v are homogeneous coordinates in P1, we can write l0 as follows x = at, y = bt, u = a, v = b | t ∈ A1\ {0} .
These equations make sense also for t = 0, and give the closure l0 of l in Bl 0(A2).
Now l0 meets ϕ−1
(0) in the point q = [u : v] ∈ P1, so we see that sending l to q gives one-to-one correspondence between lines through 0 in A2 and points of ϕ−1(0). (4) Let p = (x0, y0) ∈ A2 \ 0 , define ψ(p) = (x0, y0), [x0 : y0]
∈ Bl0(A2). Then,
ψ : A2\{0} → Bl0(A2)\ϕ−1(0) is an isomorphism which is the inverse of the restriction
of ϕ to Bl0(A2) \ ϕ−1(0).
(5) Bl0(A2) is the union of Bl0(A2) \ ϕ−1(0) and ϕ−1(0). The first piece is isomorphic to
A2 \ 0 , hence irreducible. On the other hand, we have just seen that every point of ϕ−1(0) is in the closure of some subset (the line l0) of Bl0(A2) \ ϕ−1(0). Hence,
Bl0(A2) \ ϕ−1(0) is dense in Bl0(A2), and Bl0(A2) is irreducible.
Definition 1.19. If Y is a closed subvariety of A2 passing through 0, we define the blowing-up of Y at 0 to be ˜Y = ϕ−1 Y \ 0 , where ϕ : Bl
0(A2) → A2 is the blowing-up of A2
at the point 0 described above. We denote also by ϕ : ˜Y → Y the morphism obtained by restricting ϕ : Bl0(A2) → A2 to ˜Y .
Remark 1.20. Note that ϕ induces an isomorphism of ˜Y \ ϕ−1(0) to Y \ 0 , so that ϕ is a birational morphism of ˜Y to Y .
Remark 1.21. To blow up any other point p of A2, make a linear change of coordinates
sending p to 0.
Definition 1.22. Let ϕ : Bl0(A2) → A2 be the blowing-up of A2 at 0 as in Definition 1.17.
Then, we can write Bl0(A2) = A2x1,y1 ∪ A 2 x2,y2 where A2x1,y1 = (x1, x1y1), [1 : y1] ⊂ Bl0(A2), A2x2,y2 = (x2y2, y2), [x2 : 1] ⊂ Bl0(A2)
are called respectively the first and the second chart of the blowing-up. The restriction of ϕ to the first chart A2x1,y1 is given by
A2x1,y1 −→ A 2
x,y, (x1, x1y1), [1 : y1] 7−→ (x1, x1y1),
while the restriction of ϕ to the second chart A2x2,y2 is given by
A2x2,y2 −→ A 2
x,y, (x2y2, y2), [x2 : 1] 7−→ (x2y2, y2).
Note that ϕ−1(0) ' P1 is locally defined by x
1 = 0 in the first chart A2x1,y1 and by y2 = 0 in
the second chart A2 x2,y2.
Remark 1.23. Let Y be an affine curve in A2 defined by the equation f (x, y) = 0 and let m = mult0(Y ) be the multiplicity of the curve Y at 0. Then, the strict transform ˜Y of Y is
locally defined in the first chart A2x1,y1 by
f (x1, x1y1)
xm 1
= 0 and in the second chart A2
x2,y2 by
f (x2y2, y2)
y2m = 0.
Definition 1.24. Let S be a surface and p ∈ S. Then, there exist a surface ˜S and a morphism π : ˜S → S, which are unique up to isomorphisms, such that
(i) the restriction of π to π−1 S \p is an isomorphism onto S \ p ; (ii) E := π−1(p), is isomorphic to P1.
We shall say that π is the blowing-up of S at p and E is the exceptional curve of π.
Take a neighbourhood U of p on which there exist local coordinates x, y at p (i.e. the curves x = 0, y = 0 intersect transversely at p). We can assume that p is the only point of U in the intersection of these two curves. Define the subvariety ˜U of U × P1 by
˜
U := (x, y), [u : v] ∈ U × P1 | xv = uy .
It is clear that the projection π : ˜U → U is an isomorphism over the points of U where at most one of the coordinates x, y vanishes, while π−1(p) = {p} × P1. We get S by passing ˜U
and S \ {p} along U \ {p} ∼= ˜U \ π−1(p).
Definition 1.25. Let C be an irreducible curve on S. The closure of π−1 C \p in ˜S is an irreducible curve ˜C on ˜S, which we call the strict transform of C. Let us call π−1(C) the total inverse image of C and π∗C the total transform of C.
Remark 1.26. Note that π−1(C) coincides with ˜C if and only if p 6∈ C, otherwise π−1(C) = ˜
C ∪ E.
Proposition 1.27. Let S be a surface, π : ˜S → S the blowing-up of a point p ∈ S and E ⊂ ˜S the exceptional curve. Then,
(i) there is an isomorphism Pic S ⊕ Z → Pic ˜S defined by (C, n) 7→ π∗C + nE. Hence, Pic ˜S = π∗Pic S ⊕ ZE.
(ii) for each C, D ∈ Pic S, one has π∗C.π∗D = C.D. Moreover, E.π∗C = 0 and E2 = −1.
(iii) KS˜ = π∗KS+ E.
Proof. See Lemma II.3 in [3].
Lemma 1.28. Let π be as above and let C be an irreducible curve on S. Setting m > 0 the multiplicity of C at p, one has π∗C = ˜C + mE, ˜C.E = m and ˜C2 = C2− m.
1.2.2
A sequence of blowing-ups of points
Definition 1.29. Let p1 ∈ P2 = S0 be a point. Consider the blowing-up π1 : S1 → P2 at p1
and denote by E11 = π1−1(p1) the exceptional curve.
Let p2 ∈ S1 and π2 : S2 → S1 be the blowing-up of S1 at p2. We denote the exceptional curve
by E2
2 and the strict transform of E11 in S2 by E12. One observes that if p2 6∈ E11, then the
total transform of E11 in S2 coincides with the strict transform E12. Otherwise, if p2 ∈ E11,
by Remark 1.26 and Lemma 1.28, it follows: (π1◦ π2)−1(p1) = π−12 (E 1 1) = E 2 1 ∪ E 2 2 and π ∗ 2(E 1 1) = E 2 1 + E 2 2.
Repeating the construction r times, one defines for all i = 1, . . . , r: • the blowing-up πi : Si → Si−1 of Si−1 at pi ∈ Si−1;
• the exceptional curve Ei i = π
−1
i (pi) of Si;
• for any j > i, πij : Sj → Si−1 the composition πi◦ πi+1◦ . . . ◦ πj;
• the total transform E∗
i = π∗i+1,r(Eii) of Eii in S = Sr;
• for any j > i, the strict transform Eij of Eii in Sj;
• the strict transform Ei := Eir of Eii in S;
• ,i and , respectively the intersection number in Si and in S.
All these data form the sequence of blowing-ups π = π1r : S = Sr
πr
→ Sr−1 → . . . → S1 π1
→ S0 = P2
at the points p1, . . . , pr. From now on, with abuse of notation, we say that Ei and Ei∗ are
respectively the strict and the total transform of the point pi in S.
Remark 1.30. Note that the strict transform Eij for any j > i can be defined inductively:
Eij = π∗j(Eij−1) if pj 6∈ Eij−1, π∗j(Eij−1) − Ejj if pj ∈ Eij−1.
Lemma 1.31 (cf. [7, Lem 1.1.8, Chap 1]). Let π : S → P2 be a sequence of blowing-ups of
r points, as above. Then, one has
Pic S ∼= Pic P2⊕ Zr,
where Pic P2 ,→ Pic S is defined by C 7→ π∗(C) and Ei∗ 1≤i≤r is a set of generators of Zr. The intersection numbers of the Ei∗ are
Ei∗, Ej∗ = −δij = −1 if i = j, 0 otherwise.
Proof. The first part of assertion follows by induction on r and Proposition 1.27.
As for the second part, by definition of Ei∗ and by part (ii) of Proposition 1.27, one has Ei∗, Ei∗ = πi+1,r∗ (Eii), πi+1,r∗ (Eii) = Eii, Eiii = −1.
Similarly, if j > i, one has
Ei∗, Ej∗ = πj+1,r∗ (πi+1,j∗ (Eii)), π∗j+1,r(Ejj) = πi+1,j∗ (Eii), Ejjj = 0.
Remark 1.32 (see e.g. [7, §1.3.7]). One can see that another set of generators of Zr in the previous lemma is {Ei}1≤i≤r. Moreover, the basis change matrices N = (nij) and M =
(mij) = N−1, such that Ei = r X j=1 nijEj∗, E ∗ i = r X j=1 mijEj
are given by N = Ir− Q where Ir is the (r × r) identity matrix and Q = (qij) is defined by
qij = 1 if pj ∈ Eij−1, 0 otherwise. In Chapter 2, QT will be called the proximity matrix of π.
Blowing-ups of points are so important because any birational map between surfaces factors through blowing-ups in the following sense:
Theorem 1.33. Let ϕ : X 99K Y be a birational map between surfaces. Then, there is a surface Z and birational morphisms πX : Z → X and πY : Z → Y , which are sequences of
blowing-ups of points, such that the following diagram commutes:
X ϕ Y.
Z
πX πY
For the proof see e.g. Theorem 4.9, §3.3, Chapter 4 in [25]. In particular, the theorem is a corollary of the following two results:
• Let X be a surface and ϕ : X 99K Pn a rational map. Then, there exists a sequence
of blowing-ups of points of surfaces Xm πm
→ . . . π2
→ X1 π1
→ X such that the composite rational map ψ = ϕ ◦ π1◦ . . . ◦ πm : Xm → Pn is morphism.
• Let ϕ : X → Y be a birational morphism between surfaces. Then, there exists a sequence of blowing-ups of points πi : Yi → Yi−1 for i = 1, . . . , r where Y0 = Y, Yr = X
such that ϕ = π1◦ . . . ◦ πr. In other words, any birational morphism between surfaces
1.2.3
Bubble space of P
2Definition 1.34 (cf. [15, §7.3.2]). We denote by B(P2) the so-called bubble space of P2, which is defined as follows. Consider all surfaces X above P2, i.e. all surfaces X such that there exists a birational morphism X → P2. If X1, X2 are two surfaces above P2, say π1: X1 → P2
and π2: X2 → P2 are birational morphisms, one identifies p1 ∈ X1 with p2 ∈ X2 if the
birational map (π2)−1π1: X1 99K X2 is a local isomorphism at p1, that sends p1 to p2. The
bubble space B(P2) is the union of all points of all surfaces above P2 modulo the equivalence
relation generated by these identifications.
For any birational morphism X → P2, there is an injective map X → B(P2), therefore we will identify points of X with their images in B(P2).
One says that p1 ∈ B(P2) is infinitely near p2 ∈ B(P2), say p1 ∈ X1 and p2 ∈ X2, with
birational morphisms π1: X1 → P2 and π2: X2 → P2, if the birational map (π2)−1π1: X1 99K
X2 is defined at p1, sends p1 to p2, but is not a local isomorphism at p1. In such a case we
write that p1 p2.
One moreover says that p1 is in the first neighbourhood of p2, or that p1 is infinitely near
p2 of the first order, if (π2)−1π1 corresponds locally to the blow-up of p2. In such a case we
write that p1 1 p2.
If p1 p2 then one can define the infinitesimal order of p1 with respect to p2 by induction,
namely if p1 1 p3 and p3 k p2 for some k, then p1 is infinitely near p2 of order k + 1.
If p1 p2 and p1 ∈ X1, then there is a unique irreducible curve E2 ⊂ X1 which corresponds
to the exceptional curve of the blowing-up of p2 ∈ X2. One says that p1 is proximate to p2
if p1 ∈ E2. In such a case we write that p1 99K p2. Clearly, if p1 1 p2, then p1 99K p2, but
the converse is not always true.
If p1 99K p2 and p1 k p2 with k > 1, then we say that p1 is satellite to p2 and we write
p1 p2. Otherwise, if p1 is not satellite to p2, then we denote by p1 6 p2.
One says that a point p ∈ P2 ⊂ B(P2) is a proper point of P2.
Remark 1.35. Each point of B(P2) \ P2 is infinitely near a unique point of P2.
Remark 1.36. If p1 k pk, say
p1 1 p2 1 p3 1 · · · 1 pk−1 1 pk,
and p1 99K pk, then pi 99K pk also for each i = 2, . . . , k − 1.
Notation 1.37. If p1 p2 ∈ P2 where p1 ∈ X1 and π1 : X1 → P2 is a birational morphism,
we say that a plane curve C passes through p1if C passes through p2 and the strict transform
of C on X1 via π1 passes through p1.
Proposition 1.38 (Proximity inequality). Let ϕ : S → P2 be a birational morphism, that
curve and let Ci be the strict transform of C in Si for i = 1, . . . , r. Setting C0 = C and
mi = multpi(Ci−1) for i = 1, . . . , r, one has, for each j = 1, . . . , r,
mj >
X
pk99Kpj
mk.
Proof. See §2.2 in [1] or Theorem 3.5.3, Corollary 3.5.4 in [9].
1.3
Standard coordinates of infinitely near points
In this section, we want to give a way to describe infinitely near points that we call standard coordinates.
Let p1 = [a : b : c] ∈ P2. Let us consider three cases:
(i) if c 6= 0, then p1 = a c : b c : 1 = [a : b : 1];
(ii) if c = 0 and b 6= 0, then p1 =
a b : 1 : 0
= [a : 1 : 0]; (iii) if c = b = 0, then p1 = [1 : 0 : 0].
In case (i), we work on the affine chart U2 ' C2x,y, so that p1 corresponds to the point
p1 = (a, b), and we define the isomorphism α1: C2x,y → C2x0,y0 by
α1(x, y) = (x − a, y − b).
In case (ii), we work on the affine chart U1 ' C2x,z, so that p1 corresponds to the point
p1 = (a, 0), and we define the isomorphism α1: C2x,z → C2x0,y0 by
α1(x, z) = (x − a, z).
In case (iii), we work on the affine chart U0 ' C2y,z, so that p1 corresponds to the point
¯
p1 = (0, 0), and we define the isomorphism α1: C2y,z → C2x0,y0 by
α1(y, z) = (y, z).
In all three cases, we defined α1 in such a way that α1(p1) = (0, 0) ∈ C2x0,y0.
We blow-up C2
x0,y0 at (0, 0) and we consider the first chart C 2
x1,y1 where the blowing-up map
is given in coordinates by x0 = x1, y0 = x1y1, cf. Definition 1.22.
In this chart, the exceptional curve E1 has local equation x1 = 0, hence a point p2 1 p1
corresponds either to the point (0, t2) ∈ E1 with t2 ∈ C or to the point which is the origin of
the second chart. In the former case, let us say that p2 has standard coordinates p2 = (p1, t2),
while in the latter case let us say that p2 has standard coordinates p2 = (p1, ∞). Setting
Remark 1.39. Recall that a point p2 1 p1 corresponds to the direction of a line passing
through p1. More precisely, one can see that the point p2 = (p1, t2), with p1 = [a : b : c],
corresponds to the line defined by the following equation
cy − bz = t2(cx − az) when c 6= 0 and t2 ∈ C,
cx − az = 0 when c 6= 0 and t2 = ∞,
bz = t2(bx − ay) when c = 0, b 6= 0 and t2 ∈ C,
bx = ay when c = 0, b 6= 0 and t2 = ∞,
z = t2y when b = c = 0 and t2 ∈ C,
y = 0 when b = c = 0 and t2 = ∞.
In other words, the above equations define the unique line passing through p1 and p2.
We want to go on by blowing-up at p2 = (p1, t2), with t2 ∈ P1 = C ∪ {∞}. Either t2 ∈ C
or t2 = ∞. In the former case, with notation as above, let α2: C2x1,y1 → C 2 ¯
x1,¯y1 be the
isomorphism defined by
α2(x1, y1) = (x1, y1− t2).
In the latter case, p2 corresponds to the origin of the second chart of the blowing-up of C2x0,y0
at (0, 0) that we write C2
x01,y01, where the blowing-up map is given by x0 = x 0 1y 0 1, y0 = y10. Let α2: C2x0 1,y 0 1 → C 2 ¯ x1,¯y1 be the isomorphism α2(x01, y 0 1) = (y 0 1, x 0 1).
In this way, in both cases, in C2 ¯
x1,¯y1 the exceptional curve E1 has local equation ¯x1 = 0 and
the point p2 corresponds to the origin (0, 0).
We blow-up C2 ¯
x1,¯y1 at (0, 0) and we consider the first chart C 2
x2,y2 where the blowing-up map
is given in coordinates by ¯x1 = x2, ¯y1 = x2y2. In this chart, the exceptional curve E2 has
local equation x2 = 0, hence a point p3 1 p2 corresponds either to the point (0, t3) ∈ E2
with t3 ∈ C or to the point which is the origin of the second chart.
Let us say that p3 has standard coordinates p3 = (p1, t2, t3), where either t3 ∈ C in the
former case or t3 = ∞ in the latter case.
Note that the strict transform of E1 can be seen only in the second chart and it meets
E2 at the origin of the second chart. In other words, the point with standard coordinates
(p1, t2, ∞) is satellite to p1.
More generally, let us proceed by induction of the infinitesimal order. Suppose that we have blown-up the point pr−1 with standard coordinates pr−1 = (p1, t2, . . . , tr−1), with ti ∈ P1 =
C ∪ {∞}, i = 2, . . . , r − 1. Following the procedure described above, we may assume that pr−1 is the origin of a chart C2¯xr−1,¯yr−1 in such a way that the exceptional curve Er−1 has
local equation ¯xr−1 = 0.
In the first chart of the blowing up of C2 ¯
xr−1,¯yr−1 at (0, 0), given in coordinates by ¯xr−1 =
corresponds either to the point (0, tr) ∈ Er with tr ∈ C or to the point which is the origin
of the second chart, given in coordinates by ¯xr−1 = xryr, ¯yr−1 = yr.
Let us say that pr has standard coordinates pr = (p1, t2, . . . , tr), where tr ∈ C in the former
case and tr = ∞ in the latter case.
The above discussion proves the following:
Lemma 1.40. Let p1 ∈ P2. Then, there is a one-to-one correspondence between points
infinitely near p1 of order r and (P1) r
= P1× . . . × P1
| {z }
r-times
.
Corollary 1.41. There is a one-to-one correspondence between points infinitely near a proper point of order r and W = P2× (P1)r.
Definition 1.42. We call standard coordinates of an infinitely near point the point of W obtained with the above construction.
Example 1.43. Let C be the conic in P2 defined by 2xy + 3yz − z2 = 0. A point of C is
p1 = [−1 : 1 : 2]. We claim that C passes through the points with standard coordinates
p2 = p1, − 1 2 , p3 = p1, − 1 2, 1 2 , p1, − 1 2, 1 2, − 1 2 , p1, − 1 2, 1 2, − 1 2, 1 2 , and so on.
In the affine chart U2 ' C2¯x,¯y, the point p1 corresponds to the point ¯p1 = (−1/2, 1/2) and
C is locally defined by 2¯x¯y + 3¯y − 1 = 0. The isomorphism α1: Cx,¯2¯y → C2x0,y0 defined by
α1(¯x, ¯y) = (¯x + 1/2, ¯y − 1/2) is such that α1(¯p1) = (0, 0) and C is locally defined in C2x0,y0 by
2x0y0+ x0 + 2y0 = 0. (1.2)
In the first chart of the blow-up of C2
x0,y0 at (0, 0), given in coordinates by x0 = x1, y0 = x1y1,
the strict transform of C has local equation 2x1y1 + 2y1+ 1 = 0, so that it passes through
the point (0, −1/2) and we say that C passes through the point p2 1 p1 with standard
coordinates p2 = (p1, −1/2).
Let α2: C2x1,y1 → C 2 ¯
x1,¯y1 be the isomorphism α2(x1, y1) = (x1, y1 + 1/2). In the first chart
of the blow-up of C2 ¯
x1,¯y1 at (0, 0), given in coordinates by ¯x1 = x2, ¯y1 = x2y2, the strict
transform of C has local equation 2x2y2− x2 + 2y2 = 0 so that it passes through the point
(0, 1/2) and we say that C passes through the point p3 1 p2 with standard coordinates
p3 = (p1, −1/2, 1/2).
Let α3: C2x2,y2 → C 2 ¯
x2,¯y2 be the isomorphism α3(x2, y2) = (x2, y2 − 1/2). In the first chart
of the blow-up of C2x¯2,¯y2 at (0, 0), given in coordinates by ¯x2 = x3, ¯y2 = x3y3, the strict
transform of C has local equation
2x3y3+ x3+ 2y3 = 0
that is the same equation (1.2), replacing x3 with x0 and y3 with y0. It follows that the
Example 1.44. Let us denote by F (n) the n-th Fibonacci number, starting from F (0) = F (1) = 1, then F (n) = F (n − 1) + F (n − 2) for n ≥ 2.
For n ≥ 1, let Cn be the curve in P2 defined by xF (n)yF (n+1)− zF (n+2) = 0. The curve Cn
has a singular point of multiplicity F (n + 1) at p1 = [1 : 0 : 0]. We claim that Cn passes
through the points p2, p3, . . . , pn+1 with respective standard coordinates
p2 = (p1, ∞) , p3 = (p1, ∞, ∞) , p4 = (p1, ∞, ∞, ∞) , . . . , pn+1= (p1, ∞, . . . , ∞
| {z }
n times
),
with respective multiplicities F (n), F (n − 1), . . ., F (0). In particular, for n ≥ 3, one has that pn pn−2.
We prove the claim by induction on n. For n = 1, the curve C1 has equation xy2− z3 = 0,
so C1 has a cusp at p1 with cuspidal tangent the line y = 0, so the strict transform of C1
passes through p2 with standard coordinates (p1, ∞) and it passes through p3 = (p1, ∞, ∞).
Note that p3 p1.
For n ≥ 2, in the affine chart U0 ' C2y,¯¯z, the point p1 corresponds to the origin ¯p1 = (0, 0) and
Cn is locally defined by ¯yF (n+1)− ¯zF (n+2) = 0. The isomorphism α1: Cy,¯2¯z → C2x0,y0 defined
by α1(¯y, ¯z) = (¯y, ¯z) is such that α1(¯p1) = (0, 0) and Cn is locally defined in C2x0,y0 by
xF (n+1)0 − y0F (n+2)= 0.
In the second chart of the blow-up of C2x0,y0 at (0, 0), given in coordinates by x0 = x1y1, y0 =
y1, the strict transform of Cn has local equation
xF (n+1)1 − y1F (n) = 0,
so that it has multiplicity F (n) at the origin (0, 0), that is the point with standard coordinates p2 = (p1, ∞). Let α2: C2x1,y1 → C 2 ¯ x1,¯y1 be the isomorphism α2(x1, y1) = (y1, x1). In C 2 ¯
x1,¯y1, the strict
trans-form of Cn has local equation
¯
xF (n)1 − ¯y1F (n+1)= 0. In the second chart of the blow-up of C2
¯
x1,¯y1 at (0, 0), given in coordinates by ¯x1 = x2y2, ¯y1 =
y2, the strict transform of Cn has local equation
xF (n)2 − y2F (n−1)= 0,
so that it has multiplicity F (n − 1) at the origin (0, 0), that is the point with standard coordinates p3 = (p1, ∞, ∞). Let α3: C2x2,y2 → C 2 ¯ x2,¯y2 be the isomorphism α3(x2, y2) = (y2, x2). In C 2 ¯
x2,¯y2, the strict
trans-form of Cn has local equation
xF (n−1)2 − y2F (n) = 0, and we conclude by the induction hypothesis.
1.3.1
Conics and infinitely near points
Remark 1.45. If p1 ∈ P2, p3 1 p2 1 p1 and p3 p1, i.e. p3 99K p1, then there is no smooth
curve passing through p1, p2, p3 because of the proximity inequality at p1.
Lemma 1.46. If p1 ∈ P2, p3 1 p2 1 p1 and p1, p2, p3 are collinear, namely p3 lies on
the strict transform of the line passing through p1 and p2, then there is no irreducible conic
passing through p1, p2, p3.
Proof. Up to automorphisms of P2, we may assume that p
1 = [1 : 0 : 0] and p2 = (p1, 0), so
p3 is uniquely determined by p1, p2, namely p3 = (p1, 0, 0).
Suppose that C is an irreducible conic passing through p1, p2. Then, C has equation
a2y2+ a3xz + a4yz + a5z2 = 0
where a2, a3, a4, a5 ∈ C and a2, a3 6= 0 because C is irreducible.
We work in the affine chart U0 ' C2y,z and we consider the isomorphism α1: Cy,¯2¯z → C2x0,y0
defined by α1(¯y, ¯z) = (¯y, ¯z), where the conic C has local equation
a2x20+ a3y0+ a4x0y0+ a5y20 = 0.
In the first chart of the blowing-up of C2x0,y0 at the origin (0, 0), where x0 = x1, y0 = x1y1,
the strict transform of C has local equation
a2x1+ a3y1+ a4x1y1+ a5x1y12 = 0.
Note that p2 is just the origin of C2x1,y1.
Then, the strict transform of C via the blowing-up of C2x1,y1 at the origin (0, 0) has local
equation in the first chart, where x1 = x2, y1 = x2y2,
a2+ a3y2+ a4x2y2+ a5x2y22 = 0.
Note that p3 is just the origin of C2y2,z2 but the strict transform of C does not pass through
(0, 0) because a2 6= 0.
Remark 1.47. It is easy to check that if p1 ∈ P2, p3 1 p2 1 p1, p3 6 p1 and p1, p2, p3 are
not collinear, then there are irreducible conics passing through p1, p2, p3.
Remark 1.48. Note that if p1 ∈ P2, p2 1 p1, p3 1 p1 and p2 6= p3, then there is no
irreducible conic passing through p1, p2, p3.
Lemma 1.49. Let p1, p2, p3, p4 ∈ P2 and p5 1 p1 such that no three among p1, . . . , p5 are
collinear. Then, there exists a unique irreducible conic passing through p1, . . . , p5.
Proof. Up to automorphisms of P2, we may assume that p
1 = [1 : 0 : 0], p2 = [0 : 1 : 0], p3 =
[0 : 0 : 1], p4 = [1 : 1 : 1]. Then, p5 has standard coordinates p5 = (p1, t5), namely p5 is
indeed, if t5 = 0, then p5, p2, p1 would be collinear; if t5 = 1, then p5, p4, p1 would be collinear
and finally, if t5 = ∞, then p5, p3, p1 would be collinear. Then, one can check that the conic
xz − t5xy + (t5− 1)yz = 0
is the unique irreducible conic passing through p1, . . . , p5.
Lemma 1.50. Let p1, p2, p3 ∈ P2 and p5 1 p4 1 p1 such that p5 6 p1 and no three
among p1, . . . , p5 are collinear. Then, there exists a unique irreducible conic passing through
p1, . . . , p5.
Proof. Up to automorphisms of P2, we may assume that p
1 = [1 : 0 : 0], p2 = [0 : 1 : 0], p3 =
[0 : 0 : 1] and that p4 has standard coordinates p4 = (p1, 1), namely p4 is infinitely near
p1 of the first order in the direction of the line y = z. Then, p5 has standard coordinates
p5 = (p1, 1, t5), where t5 ∈ C∗: indeed, if t5 = 0 then p5, p4, p1 would be collinear and if
t5 = ∞, then p5 p1. Then, one can check that the conic
xz − xy − t5yz = 0
is the unique irreducible conic passing through p1, . . . , p5.
Lemma 1.51. Let p1, p2, p3 ∈ P2 and p4 1 p1, p5 1 p2 such that no three among p1, . . . , p5
are collinear. Then, there exists a unique irreducible conic passing through p1, . . . , p5.
Proof. Up to automorphisms of P2, we may assume that p
1 = [1 : 0 : 0], p2 = [0 : 1 : 0], p3 =
[0 : 0 : 1] and that the two lines, one through p1, p4 and the other one through p2, p5, meet
at [1 : 1 : 1], namely p4 is infinitely near p1 of the first order in the direction of the line y = z
and p5 is infinitely near p2 of the first order in the direction of the line x = z. In other words,
p4 has standard coordinates p4 = (p1, 1) and p5 has standard coordinates p5 = (p2, 1). Then,
it is clear that the conic
xy − yz − xz = 0 is the unique irreducible conic passing through p1, . . . , p5.
Lemma 1.52. Let p1, p2 ∈ P2 and p5 1 p3 1 p1, p4 1 p2 such that p5 6 p1 and no three
among p1, . . . , p5 are collinear. Then, there exists a unique irreducible conic passing through
p1, . . . , p5.
Proof. Up to automorphisms of P2, we may assume that p1 = [1 : 0 : 0], p2 = [0 : 1 : 0], and
that the two lines, one through p1, p3 and the other one through p2, p4, meet at [0 : 0 : 1],
namely p3 is infinitely near p1 of the first order in the direction of the line y = 0 and p4
is infinitely near p2 of the first order in the direction of the line x = 0. In other words, p3
has standard coordinates p3 = (p1, ∞) and p4 has standard coordinates p4 = (p2, ∞). Then,
p5 has standard coordinates p5 = (p1, ∞, t5) where t5 ∈ C∗: indeed, if t5 = 0 then p5, p3, p1
would be collinear and if t5 = ∞, then p5 p1. One can check that the conic
is the unique irreducible conic passing through p1, . . . , p5.
Remark 1.53. The previous lemmas are a more precise explanation of Remark 4.2.1 in Chapter V in [19].
Lemma 1.54. Let p1, p2 ∈ P2 and p5 1 p4 1 p3 1 p1 such that p4 6 p1, p5 6 p3 and no
three among p1, . . . , p4 are collinear. Then, there exists a unique irreducible conic passing
through p1, . . . , p5.
Proof. Up to automorphisms of P2, we may assume that p
1 = [1 : 0 : 0], p2 = [0 : 1 : 0] and
p3, p4 have standard coordinates respectively p3 = (p1, ∞) and p4 = (p1, ∞, 1), according
to the proof of the previous lemma. Then, p5 has standard coordinates p5 = (p1, ∞, 1, t5)
where t5 ∈ C: indeed, if p5 = ∞, then we would have p5 p3, contradicting the hypothesis.
One can check that the conic
xy + t5yz − z2 = 0
is the unique irreducible conic passing through p1, . . . , p5.
Lemma 1.55. Let p5 1 p4 1 p3 1 p2 1 p1 ∈ P2 such that p3 6 p1, p4 6 p2, p5 6 p3
and p1, p2, p3 are not collinear. Then, there exists a unique irreducible conic passing through
p1, . . . , p5.
Proof. Up to automorphisms of P2, we may assume that p1 = [1 : 0 : 0] and p2, p3, p4 have
standard coordinates respectively p2 = (p1, ∞), p3 = (p1, ∞, 1), p4 = (p1, ∞, 1, 0), according
to the proof of the previous lemma. Then, p5 has standard coordinates p5 = (p1, ∞, 1, 0, t5)
where t5 ∈ C: indeed, if t5 = ∞, then we would have p5 p3, contradicting the hypothesis.
One can check that the conic
xy − z2+ t5y2 = 0
is the unique irreducible conic passing through p1, . . . , p5.
1.4
Plane Cremona maps
The plane Cremona group, denoted by Cr(P2) or Bir(P2), is the group of birational maps of
the projective plane P2 into itself. Such maps can be written as the following form
ϕ : P2 99K P2, [x : y : z] 7→ [ϕ0(x, y, z) : ϕ1(x, y, z) : ϕ2(x, y, z)] (1.3)
where ϕi ∈ C[x, y, z]d for any i = 0, 1, 2 are homogeneous polynomials of the same degree
d, that is called the degree of ϕ if ϕ0, ϕ1, ϕ2 have no common factor. Usually, abusing of
notation, let us write (1.3) as ϕ = [ϕ0 : ϕ1 : ϕ2].
Plane Cremona maps of degree 1 are automorphisms of P2, i.e. elements of Aut(P2) ' PGL 3.
Plane Cremona maps of degree 2 (3, 4, resp.) are called quadratic (cubic, quartic, resp.). The elementary quadratic transformation is:
The fundamental result concerning the plane Cremona group is the following theorem: Theorem 1.56 (Noether-Castelnuovo). The group Cr(P2) is generated by Aut(P2) and σ.
Proof. See [10] or [1] for a modern reference.
Let ϕ ∈ Cr(P2) be a plane Cremona map of degree d. Then, let p
1, . . . , pn ∈ P2 be the
(proper) base points of the net (linear system of dimension 2) Λ defining ϕ. According to Theorem 1.33, there exist a surface Z and birational morphisms π1: Z → P2 and π2: Z → P2
such that π2 = ϕ ◦ π1. The birational morphism π1: Z → P2 is the sequence of blowing-up
maps at points p1, . . . , pn and pn+1, . . . , pr ∈ B(P2) as in Section 1.2. Denote by m1, . . . , mr
the multiplicities of p1, . . . , pr of the net Λ, namely the multiplicities at p1, . . . , pr (of the
strict transform) of a general curve of the net Λ. With a little abuse of notation, let us say that p1, . . . , pr are the base points of ϕ with respective multiplicities m1, . . . , mr, and let us
write mi = multpi(ϕ) for i = 1, . . . , r. Then, it is classically known that (see e.g [1, §2.5]),
d2− 1 = r X i=1 m2i, 3(d − 1) = r X i=1 mi, (1.5)
and (d; m1, . . . , mr) is called the characteristic of ϕ.
Recall that not all solutions (d; m1, . . . , mr) of conditions (1.5) are characteristic of a plane
Cremona map (see e.g [1, §5.2]).
Definition 1.57. A plane Cremona map ϕ is called de Jonqui`eres if it has degree d and a base point of multiplicity d − 1.
Equations (1.5) imply that plane Cremona maps of degree 2 and 3 are de Jonqui`eres. Definition 1.58. A plane Cremona map ϕ is called involutory, or an involution, if ϕ−1 = ϕ. Definition 1.59. Let us say that two plane Cremona maps ϕ, ϕ0: P2 99K P2 are equivalent
if there exist two automorphisms α, α0 ∈ Aut(P2) such that
ϕ0 = α0◦ ϕ ◦ α.
Remark 1.60. The automorphism α0 changes the basis of the homaloidal net defining ϕ, while α changes the position of the base points of the map. In particular, two plane Cremona maps defined by the same homaloidal net are equivalent.
1.4.1
Quadratic plane Cremona maps
We have already defined the elementay quadratic transformation σ in (1.4). The map σ is clearly an involution and it has the coordinate points as base points of multiplicity 1. Definition 1.61. Let us say that a quadratic plane Cremona map ϕ is ordinary if ϕ has three proper base points.
Remark 1.62. Let p1, p2, p3 be the proper base points of an ordinary quadratic map ϕ. Since
there exists an automorphism α : P2 → P2 that maps p
1, p2, p3 to the coordinate points, it
follows that ϕ is equivalent to σ.
On the other hand, a plane Cremona map equivalent to σ is clearly ordinary quadratic. Remark 1.63. For each α ∈ Aut(P2), the map α−1◦ σ ◦ α is involutory ordinary quadratic,
but not all involutory ordinary quadratic maps have this form, like e.g. the map ϕ = [yz : xy : xz], cf. [24].
There are other two fundamental quadratic maps, which are not ordinary. Example 1.64. The quadratic map
ρ : P2 99K P2, [x : y : z] 7→ [xy : z2 : yz], (1.6)
is an involution which is not ordinary, namely ρ has two proper base points p1 = [1 : 0 : 0],
p2 = [0 : 1 : 0] and the third base point p3 is the point infinitely near p1 with standard
coordinates p3 = (p1, ∞), that is the point in the direction of the line y = 0.
Example 1.65. The quadratic map
τ : P2 99K P2, [x : y : z] 7→ [x2 : xy : y2− xz], (1.7) is an involution which has only one proper base point, that is p1 = [0 : 0 : 1], while the
other two base points p2 and p3 are infinitely near p1 and they have standard coordinates
respectively p2 = (p1, ∞) and p3 = (p1, ∞, 1).
Remark 1.66. It is classical well-known that any quadratic plane Cremona map is equivalent to one and only one among σ, ρ and τ .
More generally, one can see that the set of quadratic plane Cremona maps has a natural structure of quasi-projective variety of dimension 14 in P17, whose properties have been
ex-tensively studied by Cerveau and D´eserti in [11].
Definition 1.67. Let us say that a quadratic plane Cremona map ϕ is • of the second type if ϕ is equivalent to ρ;
• of the third type if ϕ is equivalent to τ .
In the next sections, we will need to construct examples of quadratic plane Cremona maps with some given property. Let us now see some of these constructions.
Example 1.68. Let p0, p1, p2 be three non-collinear points in P2. An involutory ordinary
`2 `1
`0
p0
p1 p2
Suppose that the coordinates of p0, p1, p2are
respectively [a1 : a2 : a3], [b1 : b2 : b3] and
[c1 : c2 : c3]. Let α be the automorphism of
P2 associated to the matrix
M = a1 b1 c1 a2 b2 c2 a3 b3 c3 ∈ PGL3, (1.8)
where det(M ) 6= 0 because p0, p1, p2 are not
aligned. Then, the plane Cremona map ϕ defined by ϕ = α◦σ ◦α−1 is an ordinary, involutory quadratic map based at p0, p1, p2.
Example 1.69. Let p0, p1 be two distinct points in P2 and let p2 be infinitely near p0 in the
direction of a line `, not passing through p1. An involutory quadratic plane Cremona map
based at p0, p1, p2 can be constructed as follows.
`
p0
p1 q
Suppose that the coordinates of p0, p1 are
re-spectively [a1 : a2 : a3], [b1 : b2 : b3]. Choose
a point q = [c1 : c2 : c3] on ` different from
p0. Let α be the automorphism of P2
asso-ciated to the matrix M as in (1.8), that has det(M ) 6= 0 because p0, p1, q are not aligned.
Then, the plane Cremona map ϕ defined by ϕ = α ◦ ρ ◦ α−1 is an involutory quadratic map based at p0, p1, p2.
We need to know the behaviour of plane Cremona maps under the composition with ordinary quadratic maps. A first result is the following classical proposition.
Proposition 1.70. Let p1, p2, p3 be the base points of an involutory ordinary quadratic plane
Cremona map % : P2 99K P2. Let ϕ : P2 99K P2 be a plane Cremona map of degree d > 1 with base points p4, . . . , pr and possibly p1, p2, p3. Denote by mi the multiplicity of ϕ at pi,
i = 1, . . . , r (that is mi = 0 if pi is not a base point of ϕ, i = 1, 2, 3). Suppose, moreover,
that p4, . . . , pr are proper points not lying on the triangle with vertices p1, p2, p3.
Then, the composite map ϕ ◦ %−1 = ϕ ◦ % has degree d − ε, where ε = m1+ m2+ m3− d,
and it has %(pi), i = 4, . . . , r, as base points of multiplicity mi. Furthermore, it has
multi-plicity mi− ε ≥ 0 at pi, i = 1, 2, 3 (that is, pi is not a base point of ϕ ◦ % when ε = mi).
Proof. See, e.g., Corollary 4.2.6 in [1].
Proposition 1.71. Let p1, p2, p3 be the base points of a quadratic plane Cremona map
p4, . . . , pr and possibly p1, p2, p3. Denote by mi the multiplicity of ϕ at pi, i = 1, . . . , r (that
is mi = 0 if pi is not a base point of ϕ, i = 1, 2, 3).
Then, the composite map ϕ ◦ %−1 has degree d − ε, where ε = m1+ m2+ m3− d.
Proof. See, e.g., Proposition 4.2.5 in [1].
We will later see what happens when the base points of ϕ are either infinitely near or belonging to the triangle with vertices p1, p2, p3.
Chapter 2
Weighted proximity graphs of plane
Cremona maps
In this chapter we first recall the definition and the main properties of the proximity matrices and the admissible oriented graphs which encode sequences of blowing-ups. We then define the weighted proximity graph of a given plane Cremona map, starting from the proximity properties of the base points of the Cremona map. For small degree maps, we finally intro-duce the enriched weighted proximity graph that we will use to classify equivalence classes of plane Cremona maps.
2.1
Admissible digraphs
For notation and definitions about directed graphs, see e.g. [2]. For more properties of admissible graphs, we refer to [7, Chap. 1].
Definition 2.1. A directed graph, or briefly digraph, G is a pair G = (V, F ) where V is a finite set of elements, called vertices, and F is a set of ordered pairs of distinct elements of V . An element (u, v) ∈ F where u, v ∈ V is denoted by u → v, and it is called an arc, or an arrow, from u to v.
Remark 2.2. According to Definition 2.1, a digraph has no loop, i.e. an arrow u → u where u is a vertex, and it has no multiple arcs between the same vertices.
Definition 2.3. Let G = (V, F ) be a digraph. Then the external degree and internal degree of a vertex v of G are respectively defined as follows:
outdeg(v) = ]u ∈ V
v → u , indeg(v) = ]u ∈ V
u → v .
Definition 2.4. Let G = (V, F ) be a digraph. Choose a bijection ψ : 1, . . . , n → V , where n = ]V is the number of vertices of G. Then the (n × n)-matrix AG = (aij) defined
by aij = 1 if ψ(i) → ψ(j), 0 otherwise
is called the adjacency matrix of G with respect to ψ.
Definition 2.5. A digraph G = (V, F ) is called acyclic if it has no cycle.
Remark 2.6. Let G = (V, F ) be an acyclic digraph. Then, G has at least one vertex of external degree 0, see Proposition 1.4.2 in [2].
Remark 2.7. Let G = (V, F ) be an acyclic digraph. Then, there exists an ordering of the vertices of G such that the adjacency matrix AG is a strictly lower triangular matrix, see
Proposition 1.4.3 in [2].
Definition 2.8. Two digraphs G = (V, F ) and G0 = (V0, F0) are isomorphic if there exists a bijection φ : V → V0 such that for any u, v ∈ V :
(u, v) ∈ F ⇐⇒ (φ(u), φ(v)) ∈ F0, that is, u → v ⇐⇒ φ(u) → φ(v).
Definition 2.9. Let us say that a digraph G = (V, F ) is admissible if it is acyclic and satisfies the following three properties:
(i) each vertex has the external degree at most two;
(ii) if outdeg(u) = 2, say u → v and u → w, then either v → w or w → v;
(iii) fixing two vertices v and w, then there exists at most one vertex u such that u → v and u → w.
Remark 2.10. By Property (ii), each vertex u of external degree 2 is the vertex of a triangle as in Figure 2.1.(a), up to isomorphisms.
(a) u w v (b) u w v t
Figure 2.1: (a) Admissible triangle and (b) non-admissible quadrilateral.
Remark 2.11. Property (iii) implies that the quadrilateral of Figure 2.1.(b) is not admissi-ble. In fact there are only two types of admissible quadrilaterals, up to isomorphisms, shown in Figure 2.2.
u w v t u w v t
Figure 2.2: Admissible quadrilaterals.
Lemma 2.12. An admissible, connected, digraph G has exactly one vertex with external degree 0.
Proof. By Remark 2.6, there is at least a vertex of external degree 0. Suppose that we have two vertices u and v of external degree 0. Since G is connected, then there exists a (non-directed) path starting at u and ending at v and we can choose one such path of minimum length k. Note that k ≥ 2 since u → v and v → u are not possible, because outdeg(u) = outdeg(v) = 0. Denote such path from u to v by {u = u0, u1}, {u1, u2}, . . . , {uk−1, v = uk}.
We claim that there exists a vertex uj, 1 6 j < k, of external degree 2 for G in the path,
that means that
uj → uj−1, uj → uj+1.
In fact, we know that u1 → u0, since u0 = u has external degree 0. If u1 → u2, then u1 is
the vertex we are looking for, otherwise we consider the path starting from u1 and ending
to uk= v. Our claim follows by induction on the length of the path. Then by property (ii)
of Definition 2.9, there exists an arrow either uj−1 → uj+1 or uj+1→ uj−1, so there exists a
path that connects u and v with k − 1 edges, a contradiction with the assumption that k is minimal.
Corollary 2.13. The number of connected components of an admissible digraph is equal to the number of vertices with external degree 0.
2.2
Proximity matrices
The main reference for this section is [7, Chap. 1].
Let π : S → P2 be a birational morphism. As we saw in Section 1.2.2 of Chapter 1, the
morphism π is the composition of finitely many of blowing-ups at single points. Denote by p1, . . . , pr ∈ B(P2) the blown-up points, so that π = π1◦π2◦. . .◦πrwhere πiis the blowing-up
at the point pi for each i = 1, . . . , r.
Definition 2.14. Let us associate to a birational morphism π : S → P2 a digraph Gπ with
r vertices p1, . . . , pr and there is an arrow pi → pj if and only if pi is proximate to pj.
Q = (qij) = AGπ of the digraph Gπ is defined by qij = 1 if pi ∈ Eji−1, 0 if pi 6∈ Eji−1
and we call Q the proximity matrix associated with the birational morphism π, or simply proximity matrix of π.
Remark 2.16. In [1], the notion of proximity matrix of a cluster is different.
Remark 2.17. Note that the order of the blowing-up points is important in the definition of the proximity matrix.
Example 2.18. Let us blow up p1 = [1 : 0 : 0], p2 = [0 : 1 : 0] and p3 1 p1 with standard
coordinate p3 = (p1, ∞), i.e. p3 corresponds to the line y = 0, either in the order p1, p2, p3
or p1, p3, p2. Accordingly, we find isomorphic surfaces S and S0 and birational morphisms
π : S → P2 and π0 : S0
→ P2 with π0 = π ◦ i where i : S0 → S is the isomorphism.
The digraphs Gπ and Gπ0 are the same, up to isomorphisms, but the respective proximity
matrices Q = AGπ and Q 0 = A Gπ0 are different: Q = 0 0 0 0 0 0 1 0 0 , Q 0 = 0 0 0 1 0 0 0 0 0 .
Note that Q0 = P QP where P = P−1 is the permutation matrix
P = 1 0 0 0 0 1 0 1 0 .
Let us recall the properties of a proximity matrix:
Proposition 2.19. Let Q be the proximity matrix of a birational morphism π : S → P2. Then
(1) Q is a strictly lower triangular matrix; (2) all entries of Q are either 0 or 1;
(3) in each row of Q, there are at most two non-zero entries; (4) no row with two non-zero entries is repeated;
Proof. Properties (1) and (2) are obvious while Property (3) follows from the fact that a point can belong to at most two strict transforms of distinct exceptional curves. Using notation of Section 1.2.2 in Chapter 1, one observes that
pi 6∈ Eji−1=⇒ E i j∩ E i i = ∅ =⇒ E k j ∩ E k i = ∅ for k > i. Therefore, if pk ∈ Ejk−1∩ E k−1
i for k > i > j, then pi ∈ Eji−1, that is Property (5). Moreover,
after blowing-up pk = Eik−1 ∩ E k−1
j , it follows that Eik ∩ Ejk = ∅, so that there is no other
row with the same two non-zero entries, that is Property (4).
Lemma 2.20. In the previous proposition, Properties (4) and (5) can be replaced by the following formula
qij ≥
X
k
qkiqkj, for i > j. (2.1)
Proof. Suppose that (4) and (5) hold. Then, the sum in Formula (2.1) is either 0 or 1. If it is 0, then Formula (2.1) is trivially verified, otherwise, if it is 1, Property (5) implies that qij = 1 and Formula (2.1) holds. Vice versa, suppose that Formula (2.1) holds. If
qkj = qki = 1 with i > j, then Formula (2.1) implies qij > 1, that is qij = 1 by Property (2).
So Property (5) holds. Suppose that Property (4) fails, that means there are two different rows with the same two non-zero entries. Then Formula (2.1) implies qij > 2, a contradiction
with Property (2).
Remark 2.21. We now list other properties of the proximity matrix Q associated to a birational morphism π, which is the composition of the blowing-up at points p1, . . . , pr ∈
B(P2):
• if pi 1 pj, then qij = 1;
• the i-th row of Q is zero if and only if pi ∈ P2;
• if Ei∩ Ej 6= ∅ and i > j, then qij = 1;
• if qij = 1 and Ei∩ Ej = ∅, then there exists k > i such that qkj = qki = 1;
• pk is satellite if and only if the k-th row of Q has two non-zero entries;
• if qki = qkj = 1 with i > j, then pk pj.
Remark 2.22. Let pk pj, namely pk is satellite to pj. Then, pk n pj with n > 2, i.e.
there exists pj1, . . . , pjn−1, such that
pk 1 pjn−1 1 . . . 1 pj2 1 pj1 1 pj.
Note that pji pj for each i = 2, . . . , n − 1. Indeed, with notation of Section 1.2.2 in Chapter
1, one has
pk pj ⇐⇒ pk 99K pj ⇐⇒ pk∈ Ejk−1
that implies that pji ∈ E ji−1
